The main point of the axioms is to ensure that a well defined notion of infinitesimal spaces exists in the topos, whose existence concretely and usefully formalizes the wide-spread but often vague intuition about the role of infinitesimals in differential geometry.

This way, in smooth toposes it is possible to give precise well-defined meaning to many of the familiar computations – wide-spread in particular in the physics literature – that compute with supposedly “infinitesimal” quantities.

“The reason why I have postponed for so long these investigations, which are basic to my other work in this field, is essentially the following. I found these theories originally by synthetic considerations. But I soon realized that, as expedient ( zweckmässig ) the synthetic method is for discovery, as difficult it is to give a clear exposition on synthetic investigations, which deal with objects that till now have almost exclusively been considered analytically. After long vacillations, I have decided to use a half synthetic, half analytic form. I hope my work will serve to bring justification to the synthetic method besides the analytical one.” (Sophus Lie, Allgemeine Theorie der partiellen Differentialgleichungen erster Ordnung, Math. Ann. 9 (1876).)

Synthetic differential geometry provides this formalized language.

Remark

Another advocate of the use of infinitesimals in the late 19th century was the American philosopher Charles Sanders Peirce who also foresaw the role of non-classical logic in such a putative infinitesimal calculus:

The illumination of the subject by a strict notation for the logic of relatives had shown me clearly and evidently that the idea of an infinitesimal involves no contradiction…As a mathematician, I prefer the method of infinitesimals to that of limits, as far easier and less infested with snares. Charles Sanders Peirce, The Law of Mind, The Monist 2 (1892)

Axiomatics

The axioms of synthetic differential geometry demand that the toposEE of smooth spaces is

In his work he particularly makes use of the fact that as sophisticated as a smooth topos may be when explicitly constructed (see the section on models), being a topos means that one can reason inside it almost literally as in Set. Using this Kock’s work gives descriptions of synthetic differential geometry which are entirely intuitive and have no esoteric topos-theoretic flavor. All he needs is the assumption that the Kock-Lawvere axiom is satisfied for “numbers”. Here “numbers” is really to be interpreted in the topos, but if one just accepts that they satisfy the KL axiom, one may work with infinitesimals in this context in essentially precisely the naive way, with the topos theory in the background just ensuring that everything makes good sense.

Models

Being axiomatic, reasoning in synthetic differential geometry applies in every model for the axioms, i.e. in every concrete choice of smooth toposTT.

However, with a a sufficiently general perspective on higher geometry one finds that algebraic geometry and synthetic differential geometry are both special cases of a more general notion of theories of generalized spaces. For more on this see generalized scheme.

A standard model for well adapted synthetic toposes is obtained in terms of sheaves on duals of “germ determined” C∞C^\infty-rings. This is described in great detail in the textbook Models for Smooth Infinitesimal Analysis.

The conception and discussion of these well adapted toposes goes back to Eduardo Dubuc, who studied them in a long series of articles. He asks people to refer to this topos as the Dubuc topos.

This theory of well-adapted models was later summarized and extended in the standard textbook

Differential equation

For instance the ordinary first order homogeneous differential equation that asks the derivative of a function f:X→Af : X \to A along a vector fieldv:D→XXv : D \to X^X to be given by a specified map α:X→TA\alpha: X \to T A is given by a diagram of the form

The first model for the axioms presented there served to demonstrate that the theory is non-empty, but was hard to work with. Much of the later work was concerned with refining the model-building. For instance

develop in great detail the theory of differential geometry using the axioms of synthetic differential geometry. The main goal in these books is to demonstrate how these axioms lead to a very elegant, very intuitive and very comprehensive conception of differential geometry. Accordingly, concrete models (whose explicit description is typically much more evolved than the nice axiomatics that holds once they have been constructed) play a minor role in these books.